A gearbox output shaft takes 55 kW at 1000 rpm from a helical wheel and passes the torque out through a coupling. It is stepped for its seats, keyed for the wheel and the coupling, grooved for a retaining ring and a relief, and runs on a 6212 C3 locating bearing and an NU 212 floating bearing. This tutorial opens the Drive shaft with gear and coupling example, reads its results module by module and finds which check limits the design.

The numbers are those of this version of Axomek. Open the Drive shaft with gear and coupling example to follow along.

The model

Item	Values
Material	42CrMo4 +QT; roughness Rz 6.3 µm
Sections	Ø45 × 80 (coupling seat), Ø55 × 40 (seal), Ø60 × 26 (6212 seat), Ø70 × 14 (collar), Ø65 × 74 (gear seat), Ø62 × 116, Ø60 × 32 (NU 212 seat), Ø50 × 38 mm; 420 mm in all
B1 "Locating bearing"	x = 135 mm, 6212 C3, shaft k5, housing JS6
B2 "Floating bearing"	x = 363 mm, NU 212, shaft m5, housing H7
L1 "Helical gear"	x = 197 mm; Fx −1563 N, Fy −2198 N, Fz 5833 N; Mxy 140 670 N·mm; torque 525 000 N·mm; 12 kg
L2 "Coupling"	x = 40 mm; takes the balancing torque; 5 kg
Lubrication	Oil bath, ISO VG 220, cleanliness class 7; rings at 75 / 65 °C
Duty cycle	Nominal 60 % at 1000 rpm; Peak 5 %, loads × 1.6; Part load 35 %, loads × 0.5 at 1200 rpm
Requirements	Bearing life 30 000 h; deflection 0.05 mm, tilt 4′, von Mises 200 MPa, critical speed 3000 rpm

The gear forces follow from the torque. 55 kW at 1000 rpm is T = 525 N·m. At the pitch diameter of 180 mm the tangential force is Ft = 2T/d = 5833 N; with αn = 20° and β = 15° the radial force is Ft·tan αn / cos β = 2198 N and the axial force Ft·tan β = 1563 N. The axial force acts at the pitch radius, so it also tilts the shaft by Fa·d/2 = 140.7 N·m, the moment Mxy.

The 6212 is the locating bearing: it sits next to the coupling and takes the whole axial force. The NU 212 at the far end lets the shaft grow as it warms. Gravity is on, so the gear's and the coupling's weights add to the lateral loads.

Step 1: the shaft

Open Shaft. The banner reads All 6 checks pass.

Result	Value	Limit
Max deflection	13.5 µm	50 µm
Tilt at bearings	0.34′	4′
Max stress (von Mises)	50.8 MPa	200 MPa
1st critical speed	40 833 rpm	3000 rpm
B1 radial, axial load	4891 N, 1563 N
B2 radial load	1586 N

The shaft is stiff: the gear sits only 62 mm from B1, and the sections between the bearings are Ø60 to Ø70.

Step 2: the strength after DIN 743

On the shaft inspector's Material tab, the Strength (DIN 743) group lists the notches: the coupling keyway (x = 40 mm), the retaining ring groove (122), the bearing shoulder (146), the gear keyway (197), the hub's press-fit edge (232) and the relief groove (351), with each change of diameter as a shoulder.

At the nominal load the lowest safeties are:

Place	S fatigue	S static
Gear keyway, x = 197 mm, Ø65	10.5	20.6
Coupling keyway, x = 40 mm, Ø45	11.2	8.31
Shoulder S1/S2, x = 80 mm, Ø45	11.6	8.31

The checks take the Peak case, the gear and coupling loads × 1.6: fatigue safety 6.55 and static safety 5.19, both well above the required 1.2. At the nominal load the gear keyway governs fatigue: the bending moment is largest there and the keyway's notch factor is high. The Ø45 coupling seat governs the static safety, as the full torque passes through the smallest diameter.

Step 3: the bearings over the duty cycle

Open Bearings. Each bearing is rated over the three cases by their time shares.

Bearing	Nominal case: P, L10h, Lnmh	Duty-cycle life	s0 (Peak)
BR1, 6212 C3	6188 N, 9476 h, 30 654 h	32 694 h	4.61
BR2, NU 212	1471 N, 17.2 million h, 859 million h	1 013 million h	18.2

BR1's life is the check with the least margin in the whole example: 30 000 h of 32 694 h, 92 %. The locating bearing carries most of the radial load, because the gear is close to it, and all of the axial force. Its basic life of 9476 h alone would fail; the oil bath's thick film lifts it (κ = 3.35, aISO = 3.23 at the nominal load), and the part-load case lifts the duty-cycle life above the nominal one.

The C3 clearance matters here. The k5 seat and the warmer inner ring take up 14 µm and 10 µm of its 33 µm; 9 µm is left in operation. A Normal clearance would be used up.

Step 4: the keys

Open Keys. K1 holds the coupling (Ø45 × 80 mm key in a ductile iron hub), K2 the gear (Ø65 × 70 mm).

Key	Hub safety	Key safety	Shaft safety
K1, coupling	2.30	3.10	7.79
K2, gear	13.0	5.17	14.3

The coupling's ductile iron hub is the weakest flank: its yield strength is 250 MPa against the shaft's 900 MPa.

Step 5: the gear's press fit

Open Fits. F1 is the gear seat, Ø65 H7/s6 with a Ø120 × 70 mm hub. After smoothing the interference is 16.6 to 65.6 µm, the pressure 18.9 to 74.9 MPa.

Check	Value	Limit
Slip safety	2.00	1.5
Hub yield safety	4.90	1.1
Shaft yield safety	12.0	1.1

The slip safety, at 75 % of its limit, is the second smallest margin after BR1.

Step 6: try it

Raise BR1's Minimum life to 35000 h: the life check fails at 32 694 h. A cleaner oil (a lower cleanliness class) raises eC and with it the life.
Set the Peak case's scale to 2: the shaft's safeties drop, and so does BR1's life, as the load enters it cubed even at 5 % of the time.